7 Velocity-porosity-clay models: Castagna's empirical relations for velocities
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153
3.14 Scattering attenuation
In these expressions, y is the angle between the direction of propagation and the
3-axis (axis of symmetry), and e is the crack density parameter:
"¼
N 3
3
a ¼
V
4
where N/V is the number of penny-shaped cracks of radius a per unit volume, f is the
crack porosity, and a is the crack aspect ratio.
U1 and U3 depend on the crack conditions. For dry cracks
U1 ẳ
16l ỵ 2ị
;
33l ỵ 4ị
U3 ẳ
4l ỵ 2ị
3l þ Þ
For “weak” inclusions (i.e., when =ðK 0 þ 43 0 Þ is of the order 1 and is not small
enough to be neglected)
U1 ẳ
16l ỵ 2ị
1
;
33l ỵ 4ị 1 ỵ Mị
U3 ẳ
4l ỵ 2ị 1
3l ỵ ị 1 ỵ ị
where
Mẳ
40 l ỵ 2ị
;
3l ỵ 4ị
ẳ
K 0 ỵ 43 0 ịl ỵ 2ị
l ỵ ị
with K0 and m0 equal to the bulk and shear moduli of the inclusion material. The
criterion for an inclusion to be “weak” depends on its shape, or aspect ratio a, as well
as on the relative moduli of the inclusion and matrix material. Dry cavities can be
modeled by setting the inclusion moduli to zero. Fluid-saturated cavities are simulated
by setting the inclusion shear modulus to zero. Remember that these give only the
scattering losses and do not incorporate other viscous losses caused by the pore fluid.
Hudson also gives expressions for infinitely thin fluid-filled cracks:
U1 ẳ
16l ỵ 2ị
;
33l ỵ 4ị
U3 ẳ 0
These assume no discontinuity in the normal component of crack displacements and
therefore predict no change in the compressional modulus with saturation. There is,
however, a shear displacement discontinuity and a resulting effect on shear stiffness.
This case should be used with care.
For randomly oriented cracks (isotropic distribution) the P- and S-attenuation
coefficients are given as
3
o
oa
4
1 VP5
2
2
P ẳ "
ỵ
BB
2ịU
AU
1
3
VS VP 152
2 VS5
3
Á
o
oa 1 À 2 1
S ¼ "
AU1 þ 3 BU32
VS VS 75
The fourth-power dependence on o is characteristic of Rayleigh scattering.
Seismic wave propagation
104
a=L
1000
Homogeneous
D<1
Ray theory
D=1
100
ka
154
Strong scattering
10
1
Weak scattering
∆E/E = 0.1
0.1
0.01
∆E/E < 0.1
1
10
Effective medium
100
1000
104
105
kL
Figure 3.14.2 Domains of applicability for various scattering theories.
Random heterogeneous media with spatially varying velocity c ¼ c0 ỵ c0 may be
characterized by the autocorrelation function
Nrị ẳ
hr 0 ịr 0 ỵ rịi
h2 i
where x ẳ c0 /c0 and c0 denotes the mean background velocity. For small fluctuations,
the fractional energy loss caused by scattering is given by (Aki and Richards, 1980)
8 2 k4 a3 L
E
;
for Nrị ẳ er=a
ẳ
1 ỵ 4k2 a2
E
E p 2 2
2 2
ẳ k aL1 ek a ị;
E
for Nrị ẳ er
2
=a2
These expressions are valid for small DE/E values as they are derived under the
Born approximation, which assumes that the primary incident waves are unchanged
as they propagate through the heterogeneous medium.
Aki and Richards (1980) classify scattering phenomena in terms of two dimensionless numbers ka and kL, where k ¼ 2p/l is the wavenumber, a is the characteristic scale
of the heterogeneity, and L is the path length of the primary incident wave in the
heterogeneous medium. Scattering effects are not very important for very small or very
large ka, and they become increasingly important with increasing kL. Scattering
problems may be classified on the basis of the fractional energy loss caused by
scattering, DE/E, and the wave parameter D defined by D ¼ 4L/ka2. The wave
parameter is the ratio of the first Fresnel zone to the scale length of the heterogeneity.
Ray theory is applicable when D < 1. In this case the inhomogeneities are smooth
enough to be treated as piecewise homogeneous. Effective medium theories are appropriate when ka and DE/E are small. These domains are summarized in Figure 3.14.2.
Scattering becomes complex when heterogeneity scales are comparable with the
wavelength and when the path lengths are long. Energy diffusion models are used for
long path lengths and strong scattering.
155
3.15 Waves in cylindrical rods: the resonant bar
Uses
The results described in this section can be used to estimate the seismic attenuation
caused by scattering.
Assumptions and limitations
The results described in this section have the following limitations:
formulas for spherical and ellipsoidal inclusions are limited to low pore concentrations and wavelengths much larger than the scatterer diameter;
formulas for fractional energy loss in random heterogeneous media are limited to
weak scattering.
3.15
Waves in cylindrical rods: the resonant bar
Synopsis
Time-harmonic waves propagating in the axial direction along a circular cylindrical
rod involve radial, circumferential, and axial components of displacement, ur , uy, and
uz, respectively. Motions that depend on z but are independent of y may be separated
into torsional waves involving uy only and longitudinal waves involving ur and uz.
Flexural waves consist of motions that depend on both z and y.
Torsional waves
Torsional waves involve purely circumferential displacements that are independent
of y. The dispersion relation (for free-surface boundary conditions) is of the form
(Achenbach, 1984)
saJ0 saị 2J1 saị ẳ 0
s2 ẳ
o2
k2
VS2
where Jn ðÁÞ are Bessel functions of the first kind of order n, a is the radius of the
cylindrical rod, VS is the S-wave velocity, and k is the wavenumber for torsional waves.
For practical purposes, the lowest mode of each kind of motion is important. The
lowest torsional mode consists of displacement proportional to the radius, and the
motion is a rotation of each cross-section of the cylinder about its center. The phase
velocity of the lowest torsional mode is nondispersive and is given by
rﬃﬃﬃ
Vtorsion ¼ VS ¼
where m and r are the shear modulus and density of the rod, respectively.
156
Seismic wave propagation
Longitudinal waves
Longitudinal waves are axially symmetric and have displacement components in the
axial and radial directions. The dispersion relation (for free-surface boundary conditions), known as the Pochhammer equation, is (Achenbach, 1984)
2paẵsaị2 ỵ kaị2 J1 paịJ1 saị ẵsaị2 kaị2 2 J0 paịJ1 saị
4kaị2 paịsaịJ1 paịJ0 saị ẳ 0
p2 ¼
o2
À k2
VP2
where VP is the P-wave velocity.
The phase velocity of the lowest longitudinal mode for small ka (ka ( 1) can be
expressed as
s
i
Eh
1 14 v2 kaị2 ỵ OẵKaị4
Vlong ¼
where E is the Young modulus of the cylindrical rod and n is the Poisson ratio of the
cylindrical rod.
In the limit as
p(ka)
ﬃﬃﬃﬃﬃﬃﬃﬃ ! 0, the phase velocity tends to the bar velocity or extensional
velocity VE ¼ E=: For very large ka (ka ) 1), Vlong approaches the Rayleigh
wave velocity.
Flexural waves
Flexural modes have all three displacement components – axial, radial, and circumferential and involve motion that depends on both z and y. The phase velocity of the
lowest flexural mode for small values of ka (ka ( 1) may be written as
s
1 E
Vflex ẳ
kaị ỵ Oẵkaị3
2
The phase velocity of the lowest flexural mode goes to zero as (ka) ! 0 and
approaches the Rayleigh wave velocity for large ka values.
Bar resonance
Resonant modes (or standing waves) occur when the bar length is an integer number
of half-wavelengths:
V ¼ lf ¼
2Lf
n
where V is velocity, l is wavelength, f is the resonant frequency, L is the bar length,
and n is a positive integer.
157
3.15 Waves in cylindrical rods: the resonant bar
In practice, the shear or extensional velocity is calculated from the observed
resonant frequency, most often at the fundamental mode, where n ¼ 1.
Porous, fluid-saturated rods
Biot’s theory has been used to extend Pochhammer’s method of analysis for fluidsaturated porous rods (Gardner, 1962; Berryman, 1983). The dependence of the
velocity and attenuation of longitudinal waves on the skeleton and fluid properties
is rather complicated. The motions of the solid and the fluid are partly parallel to the
axis of the cylinder and partly along the radius. The dispersion relations are obtained
from plane-wave solutions of Biot’s equations in cylindrical (r, y, z) coordinates. For
an open (unjacketed) surface boundary condition the o–kz dispersion relation is given
by (in the notation of Berryman, 1983)
a11 a12 a13
Dopen ¼ a21 a22 0 ¼ 0
a31 a32 a33
a11 ẳ
ẵC Hịkỵ2 ỵ 2fr kz2 J0 kỵ aị ỵ 2fr krỵ J1 kỵ aị=a
ỵ ị
a12 ẳ
ẵH Cỵ ịk2 2fr kz2 J0 k aị 2fr kr J1 k aị=a
ỵ ị
a13 ẳ 2fr ksr ẵksr J0 ksr aị J1 ksr aị=a
a21 ẳ
2
M Cịkỵ
J0 kỵ aị
ỵ ị
a22 ẳ
C Mỵ ịk2 J0 k aị
ỵ ị
a23 ẳ 0
a31 ẳ
2ifr kz krỵ J1 kỵ aị
ỵ ị
a32 ẳ
2ifr kz kr J1 k aị
ỵ Þ
a33 ¼ Àfr ðks2 À 2kz2 Þksr J1 ðksr aÞ=ðikz ị
2
2
ẳ kặ
kz2 ;
krặ
ksr2 ẳ ks2 kz2
ks2 ẳ o2 fl 2 =qịfr
q!
2
1
kỵ ẳ 2 b ỵ f b f ị2 ỵ 4cd
158
Seismic wave propagation
2
k
ẳ 12 b ỵ f ỵ
q!
b f ị2 ỵ 4cd
b ẳ o2 M fl Cị=
c ẳ o2 fl M qCị=
d ẳ o2 fl H Cị=
f ẳ o2 qH fl Cị=
ẳ MH C2
ặ ẳ d=kặ2 bị ẳ kặ2 f ị=c
4
K0 Kfr ị2
H ẳ Kfr ỵ fr ỵ
D Kfr ị
3
Cẳ
K0 Kfr ịK0
D Kfr ị
Mẳ
K02
D Kfr ị
D ẳ K0 ẵ1 ỵ K0 =Kfl 1ị
ẳ 1 ị0 ỵ fl
qẳ
fl iFị
o
where
Kfr , mfr ẳ effective bulk and shear moduli of rock frame: either the dry frame or the highfrequency unrelaxed “wet frame” moduli predicted by the Mavko–Jizba squirt theory
K0 ¼ bulk modulus of mineral material making up rock
Kfl ¼ effective bulk modulus of pore fluid
f ¼ porosity
r0 ¼ mineral density
rfl ¼ fluid density
a ¼ tortuosity parameter (always greater than 1)
¼ viscosity of the pore fluid
k ¼ absolute permeability of the rock
o ¼ angular frequency of the plane wave.
The viscodynamic operator F(z) incorporates the frequency dependence of viscous
drag and is defined by
Fị ẳ
1
Tị
4 1 ỵ 2iTị=
159
3.15 Waves in cylindrical rods: the resonant bar
Tị ẳ
ber0 ị ỵ ibei0 ị ei3=4 J1 ei=4 ị
ẳ
berị ỵ ibeiị
J0 ei=4 ị
ẳ o=or ị1=2 ẳ
2 1=2
oh fl
where ber( ) and bei( ) are the real and imaginary parts of the Kelvin function,
respectively, Jn( ) is a Bessel function of order n, and h is the pore-size parameter.
The pore-size parameter h depends on both the dimensions and shape of the pore
space. Stoll (1974) found that values of between 16 and 17 for the mean grain diameter
gave good agreement with experimental data from several investigators. For spherical
grains, Hovem and Ingram (1979) obtained h ¼ fd/[3(1 – f)], where d is the grain
diameter.
This dispersion relation gives the same results as Gardner (1962). When the surface
pores are closed (jacketed) the resulting dispersion relation is
a11 a12 a13
Dclosed ¼ a31 a32 a33 ¼ 0
a41 a42 a43
a41 ẳ
krỵ J1 kỵ aị
;
ỵ
a42 ẳ
kr ỵ J1 k aị
;
ỵ
a43 ẳ ksr J1 ksr aịfl =q
For open-pore surface conditions the vanishing of the fluid pressure at the surface
of the cylinder causes strong radial motion of the fluid relative to the solid. This
relative motion absorbs energy, causing greater attenuation than would be present in a
plane longitudinal wave in an extended porous saturated medium (White, 1986).
Narrow stop bands and sharp peaks in the attenuation can occur if the slow P-wave
has wavelength l < 2.6a. Such stop bands do not exist in the case of the jacketed,
closed-pore surface. A slow extensional wave propagates under jacketed boundary
conditions but not under the open-surface condition.
Uses
The results described in this section can be used to model wave propagation and
geometric dispersion in resonant bar experiments.
Assumptions and limitations
The results described in this section assume the following:
an isotropic, linear, homogeneous, and elastic–poroelastic rod of solid circular
cross-section;
for elastic rods the cylindrical surface is taken to be free of tractions; and
for porous rods, unjacketed and jacketed surface boundary conditions are assumed.
160
Seismic wave propagation
3.16
Waves in boreholes
Synopsis
Elastic wave propagation in the presence of a cylindrical fluid-filled borehole involves
different modes caused by internal refraction, constructive interference and trapping
of wave energy in the borehole. The theory of borehole wave propagation has been
described in the books by White (1983), Paillet and Cheng (1991), and Tang and
Cheng (2004), where references to the original literature may be found. The dispersion
characteristics of borehole wave modes depend strongly on the shear wave velocity of
the elastic medium surrounding the borehole. Two scenarios are usually considered:
“fast” formation when the S-wave velocity in the formation is greater than the
borehole fluid velocity and “slow” formation when the S-wave velocity in the formation
is slower than the borehole fluid velocity. Wave modes are guided by the borehole only
when the formation S-wave velocity is greater than the phase velocity of the modes;
otherwise the modes become leaky modes, radiating energy into the formation.
In fast formations, pseudo-Rayleigh modes or shear normal modes exist above
characteristic cut-off frequencies. The pseudo-Rayleigh mode is strongly dispersive
and is a combined effect of reflected waves in the fluid and critical refraction along
the borehole walls. The phase velocity of the pseudo-Rayleigh wave at the cut-off
frequency drops from the shear wave velocity of the formation and approaches the
fluid velocity at high frequencies. In slow formations pseudo-Rayleigh modes do not
exist. Stoneley waves in boreholes refer to waves along the borehole interface. At low
frequencies the Stoneley waves are referred to as tube waves. Stoneley waves exist at
all frequencies and in both fast and slow formations. “Leaky P” modes exist in slow
formations and are dominated by critical refraction of P-waves at the borehole wall.
They lose energy by conversion to shear waves. Higher-order modes include the
dipole or flexural mode and the quadrupole or screw mode (Tang and Cheng, 2004).
Isotropic elastic formation
A low-frequency (static) analysis for a thick-walled elastic tube of inner radius b and
outer radius a, with Young’s modulus E, and Poisson ratio n, containing fluid with
bulk modulus B, and density r, gives the speed of tube waves as (White, 1983)
!
1 1 À1=2
ct ¼
ỵ
B M
Mẳ
Ea2 b2 ị
2ẵ1 ỵ nịa2 ỵ b2 ị À 2nb2
For a thin-walled tube with thickness h ¼ ða À bÞ; a % b; and M % Eh=2b: For a
borehole in an infinite solid a ) b and the speed of the tube wave in the lowfrequency limit is
161
3.16 Waves in boreholes
!
1 1 1=2
ct ẳ
ỵ
B
where m is the shear modulus of the formation. For a borehole with a casing of
thickness h, inner radius b, and Young’s modulus E, the tube wave velocity (in the
low-frequency limit) is
!À1=2
1
1
ct ẳ
ỵ
B ỵ Eh=2b
The ok dispersion relation gives a more complete description of the modes. The
dispersion relation (or period equation) yields characteristic cut-off frequencies and
phase velocities as a function of frequency for the different modes (Paillet and Cheng,
1991; Tang and Cheng, 2004). For a cylindrical borehole of radius R, in an infinite,
isotropic, elastic formation with Lame´ constants l and m, and density r, and openhole boundary conditions at the interface, the dispersion relation is given by (in the
notation of Tang and Cheng, 2004)
M11
M
Do; kị ẳ 21
M31
M41
M11 ẳ
M12
M22
M32
M42
M13
M23
M33
M43
M14
M24
ẳ0
M34
M44
n
In fRị fInỵ1 fRị
R
M12 ẳ pY1 pRị
n
M13 ẳ Kn sRị
R
M14 ẳ iksY1 sRị
M21 ẳ f o2 In fRị
2p 2
Y2 pRị
M22 ẳ 2k2 2 o2 Kn pRị ỵ
R
M23 ẳ
2ns